This sounds very familiar and I would echo the comments of all those who have advised checking the ECU mapping - in my experience it is important to find someone who is familiar with your specific programme (ie OMEX) rather than a generic tuner and to use someone who does road tuning, not race cars.
The fuelling transition between start/idle and part throttle/light load is notoriously difficult to map because it may under some parameters (temperature, engine load) involve a reduction in fuelling as the throttle opens. If the drop in fuelling values is too sudden you get stalling and then surging if the ECU tries to compensate. If your car was originally tuned for the track this would explain why this part of the map is not finessed.
Good news is on an M engine there shouldn't be a cat and therefore no Lamda input to further complicate the mapping.
The cold fueling is particularly difficult as the engine does not stay cold for long enough to do the adjustments. It tends to be a matter of setting the cold running condition correction and then trying to understand just how to improve it in a short period of time. You then have to wait till everything gets cold again to see what effect your changes have made. I imagine that the major manufacturers have facilities that keep everything chilled; water, air and oil, until they can get it right.
The other thing to keep in mind is that at very small throttle openings the slightest change will increase the air flow astronomically. Let us say that the idle position has the butterfly at 1.5% of the possible angular rotation and there is an air flow of say 5 kg/h. In this condition moving the butterfly by the smallest amount, say 1 degree, will increase the airflow to 10 kg/h, and increase of 100%.
I don't know about the M engine but even if there isn't a cat my understanding is that all modern petrol engines rely on a Sonda Lamda sensor to adjust the fuel mixture. The fuel map gives a close approximate to the fuel required under various conditions but the feedback from the Sonda Lamda is used to make the fine adjustments. The problem is that the Sonda Lamda can not be used until the engine (and the sensor) reaches operating temperature (in my system coolant at 70°C) so the map and the cold running condition corrections on fuel and perhaps spark advance need to be pretty correct to get smooth cold running.
I can not imagine what it has cost Ford to develop the multi layer/multi condition mapping and very sophisticated programs for the Duratec GDI engine. Even though people like Omex will never arrive at that level of sophistication they can certainly produce a system that will work like a charm on an engine equipped with throttle bodies once it has reached operating conditions but it will never be perfect under all conditions of temperature and pressure. Our cars themselves are not perfect so we can accept something as trivial as a cold running problem to get that little bit of extra performance. That won't stop me in my quest to get it right.
I use the Omex provided logging and mapping software to monitor the engine conditions and adjust the mapping. Most of the initial changes were made by Omex technicians based on my logs, now I am doing the fine tuning myself. As I have kept all of the previous maps if I do something that does not make and improvement I can simply revert to the previous version.
PS. Sorry about the long ramble.